Related papers: Rational curves on Hermitian manifolds

We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.

We study each of the 16 types of complete intersection Calabi-Yau threefolds in projective n-space times the projective line, for various n, and prove existence of isolated rational curves of bidegree (d,0) for all positive integers d on a…

In this paper, we use Uhlenbeck-Yau's continuity method to establish the correspondence between the mean curvature positivity and the HN-positivity on holomorphic vector bundles over compact Hermitian manifolds. As its application, we get a…

Let $p: S\to S_g$ be a finite covering of an orientable closed surface of genus $g$. We prove that, for $g\geq 3$, the rational homology group $H_1(S;{\mathbb Q})$ is generated by cycles supported on simple closed curves $\gamma\subset S$…

Let X be a Calabi-Yau threefold. We show that if there exists on X a non-zero nef non-ample divisor then X contains a rational curve, provided its second Betti number is greater than 4.

We prove that the moduli spaces of rational curves of degree at most $3$ in linear sections of the Grassmannian $Gr(2,5)$ are all rational varieties. We also study their compactifications and birational geometry.

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an…

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring,…

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic…

We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free…

In this paper we consider the curves $H_{k,t}^{(p)} : y^{p^k}+y=x^{p^{kt}+1}$ over $\mathbb F_p$ and and find an exact formula for the number of $\mathbb F_{p^n}$-rational points on $H_{k,t}^{(p)}$ for all integers $n\ge 1$. We also give…

Let $X$ be a compact K\"ahler manifold. We prove that if $X$ admits a smooth Hermitian metric $\omega$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(\omega)$, then $X$ is projective and rationally connected. In…

We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.

In this note, we show that on Hopf manifold $\mathbb S^{2n-1}\times \mathbb S^1$, the non-negativity of the holomorphic bisectional curvature is not preserved along the Chern-Ricci flow.

We study the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. The argument uses spreading out to a finite field, together with an adaptation to…

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-K\"ahler manifolds. Since the K\"ahler-Ricci flow…

In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the…